VVT15 - Constant temperature gradient for a distorted mesh

Description

The purpose of this test case is to verify that the computed element temperatures closely match the theoretical results using the element center conduction method for a flat plate (A) meshed with 3-node planar elements and plate (B) meshed with 4-node planar elements.

Geometry

The geometries for cases (A) and (B) are defined using 1 m square plates with 0.5 m diameter circles in the center and bounded planes. Both cases (A) and (B) are defined as 1 mm thick. Case (A) geometry consists of two bounded planes: one plane is 1 m² less a circular area 0.5 m in diameter in the center; the second plane in case (A) is a circular area 0.5 mm in diameter located in the center of the 1 m² plane. Case (B) geometry consists of 5 bounded planes: four planes make up the four quadrants which, when combined, make an area 1 m² less a circular area 0.5 m in diameter in the center; the second plane in case (B) is a circular area 0.5 m in diameter located in the center of the 1 m² plane.

Simulation model

The finite element meshes are created using 3-node planar elements (triangles) for case A and 4-node planar elements (rectangles) for case B. The following figures show the mesh for case A and case B.

The meshed elements have the following material and physical properties:

• Thermal conductivity: k = 200 W/m·°C
• Mass density: ρ = 1 kg/m³
• Thickness: 1 mm

The following boundary conditions are applied:

• Temperature constraint on the top of the plate with a value of T _top = 100 °C.
• Temperature constraint on the bottom of the plate with a value of T_bottom = 0 °C.

The following solution options are set:

• Solution Type = Steady State
• Results Options: Select heat fluxes and temperature at nodes and elements.

The default solver parameters are selected.

Theory

For one-dimensional, steady-state conduction in a plane wall with no heat generation and constant thermal conductivity, the temperature varies linearly with x only. According to that, lines of constant temperature must be parallel to surfaces of the plane wall. The temperature distribution is given by [6] :

Results

The following tables compare the computed nodal temperatures with theoretical values. The x location for each case is determined by the x coordinate of the nodes.

Temperatures for case A

Temperatures for case B